A is diagonalizable iff there are n linearly independent eigenvectors

Dependencies:

  1. Diagonalization
  2. Linear independence
  3. Inverse of a matrix
  4. Transpose of product
  5. Full-rank square matrix is invertible
  6. A matrix is full-rank iff its rows are linearly independent

Let $A$ be an $n$ by $n$ matrix. Then $L$ is diagonalizable iff there are $n$ linearly independent eigenvectors for $A$.

Proof

\begin{align} & P^{-1} \textrm{ exists} \\ &\iff \exists Q, QP = PQ = I \\ &\iff \exists Q, P^TQ^T = Q^TP^T = I \\ &\iff (P^T)^{-1} \textrm{ exists} \\ &\iff \operatorname{rank}(P^T) = n \\ &\iff \textrm{Rows of } P^T \textrm{ are linearly independent} \\ &\iff \textrm{Columns of } P \textrm{ are linearly independent} \end{align}

$(\exists P, \exists \textrm{ diagonal } D, AP = PD) \iff$ columns of $P$ are eigenvectors of $A$.

If there are $n$ linearly independent eigenvectors, make them the columns of $P$. Then $AP = PD$ ($D$ is diagonal) and $P^{-1}$ exists, so $D = P^{-1}AP$. Therefore, $A$ is diagonalizable.

If $A$ is diagonalizable, there is a $P$ such that $P^{-1}$ exists and $AP = PD$ ($D$ is diagonal). Therefore, columns of $P$ are linearly independent and they are eigenvectors of $A$. Therefore, $A$ has $n$ linearly independent eigenvectors.

Dependency for: None

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Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/composition-of-bijections-is-a-bijection
  4. /sets-and-relations/equivalence-relation
  5. Group
  6. Ring
  7. Polynomial
  8. Integral Domain
  9. Comparing coefficients of a polynomial with disjoint variables
  10. Field
  11. Vector Space
  12. Linear independence
  13. Span
  14. Decrementing a span
  15. Linear transformation
  16. Composition of linear transformations
  17. Vector space isomorphism is an equivalence relation
  18. Semiring
  19. Matrix
  20. Stacking
  21. System of linear equations
  22. Product of stacked matrices
  23. Matrix multiplication is associative
  24. Reduced Row Echelon Form (RREF)
  25. Rows of RREF are linearly independent
  26. Transpose of product
  27. Matrices over a field form a vector space
  28. Row space
  29. Elementary row operation
  30. Every elementary row operation has a unique inverse
  31. Row equivalence of matrices
  32. Row equivalent matrices have the same row space
  33. RREF is unique
  34. Identity matrix
  35. Inverse of a matrix
  36. Inverse of product
  37. Elementary row operation is matrix pre-multiplication
  38. Row equivalence matrix
  39. Equations with row equivalent matrices have the same solution set
  40. Basis of a vector space
  41. Linearly independent set is not bigger than a span
  42. Homogeneous linear equations with more variables than equations
  43. Rank of a homogenous system of linear equations
  44. Rank of a matrix
  45. Spanning set of size dim(V) is a basis
  46. A matrix is full-rank iff its rows are linearly independent
  47. Basis of F^n
  48. Matrix of linear transformation
  49. Coordinatization over a basis
  50. Basis changer
  51. Basis change is an isomorphic linear transformation
  52. Vector spaces are isomorphic iff their dimensions are same
  53. Canonical decomposition of a linear transformation
  54. Eigenvalues and Eigenvectors
  55. Diagonalization
  56. Full-rank square matrix in RREF is the identity matrix
  57. Full-rank square matrix is invertible